In a companion paper, Ulitsky & Collins (2000) applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. Their paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy–Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient.

This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung (1996), we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms. Quantitative comparisons between the EDQNM model and the DNS show good agreement. In addition, the model is shown to give excellent estimates for the dissipation coefficient defined by Yeung (1998).

We show that the EDQNM scalar covariance spectrum for two scalars with different molecular diffusivities can be approximated by the EDQNM autocorrelation spectrum for a scalar with molecular diffusivity equal to the arithmetic mean of the first two scalars. The result is exact for the case of an isotropic scalar and is shown to be a very good approximation for the scalar with a uniform mean gradient. We then exploit this relationship to derive a simple formula for the correlation coefficient of two differentially diffusing scalars as a function of their two Schmidt numbers and the turbulent Reynolds number. A comparison of the formula with the EDQNM model shows the model predicts the correct Reynolds number scaling and qualitatively predicts the dependence on the Schmidt numbers.

To investigate the degree of local versus non-local transfer of the scalar covariance spectrum, we divided the energy spectrum into three ranges corresponding to the energy-containing eddies, the inertial range, and the dissipation range. Then, by conditioning the scalar transfer on the energy contained within each of the three ranges, we have determined that the transfer process is dominated first by local interactions (local transfer) followed by non-local interactions leading to local transfer. Non-local interactions leading to non-local transfer are found to be significant at the higher wavenumbers. This result has important implications for defining simpler spectral models that potentially can be applied to more complex engineering flows.

We derive expressions for the bulk viscosity of suspension of gas bubbles in an incompressible Newtonian liquid that exsolves volatiles. The suspension is modelled as close packed spherical cells and is represented by a single cell (‘cell model’). A cell, consisting of a gas bubble centred in a spherical shell of a volatile-bearing liquid, is subjected to decompression that is applied at the cell boundary, and the resulting dilatational boundary motion and driving pressure are obtained. The dilatational motion and the driving pressure are used to define the bulk viscosity of the cell, as if it were composed of a homogeneously compressible fluid. By definition, the bulk viscosity is the relation between changes of the driving pressure and changes in the resulting expansion strain rate. The bulk viscosity of the suspension is obtained in terms of two-phase parameters, i.e. bubble radius, gas pressure and the properties of the incompressible continuous liquid phase. The resulting bulk viscosity is highly nonlinear. At the beginning of the expansion process, when gas exsolution is efficient, the expansion rate grows exponentially while the driving pressure decreases slightly, which means that the bulk viscosity is formally negative. This negative value reflects the release of the energy stored in the supersaturated liquid and its transfer to mechanical work during exsolution. Later, when bubbles are large and the gas influx decreases significantly, the strain rate decelerates and the bulk viscosity becomes positive as expected in a dissipative system. We demonstrate that amplification of seismic waves travelling through a volcanic conduit filled with a volatile saturated magma may be attributed to the negative bulk viscosity of the compressible magma. Amplification of an expansion wave may, at some level in the conduit, damage the conduit walls and initiate the opening of a new pathway for magma eruption. We also consider the energy related to positive and negative bulk viscosities.

We consider the steady viscous/inviscid interaction of a two-dimensional, nearly separated, boundary layer with an isolated three-dimensional surface-mounted obstacle, for example in the large Reynolds number flow around the leading edge of a slender airfoil at a small angle of attack. An integro-differential equation describing the effect of the obstacle on the wall shear stress valid within the interaction regime is derived and solved numerically by means of a spectral method, which is outlined in detail. Typical solutions of this equation are presented for different values of the spanwise width B of the obstacle including the limiting cases B → 0 and B → ∞. Special emphasis is placed on the occurrence of non-uniqueness. On the main (upper) solution branch the disturbances to the flow field caused by the obstacle decay in the lateral direction. Conversely a periodic flow pattern, having no decay in the spanwise direction, was found to form on the lower solution branch. These branches are connected by a bifurcation point, which characterizes the maximum (critical) angle of attack for which a solution of the strictly plane interaction problem exists. An asymptotic investigation of the interaction equation, in the absence of any obstacle, for small deviations of this critical angle clearly reflects the observed behaviour of the numerical results corresponding to the different branches. As a result we can conclude that the primarily local interaction process breaks down in a non-local manner even in the limit of vanishing (three-dimensional local) disturbances of the flow field.

The theory of point vortices in a two-dimensional ideal fluid has a long history, but on surfaces other than the plane no method of finding periodic motions (except relative equilibria) of N vortices is known. We present one such method and find infinite families of periodic motions on surfaces possessing certain symmetries, including spheres, ellipsoids of revolution and cylinders. Our families exhibit bifurcations. N can be made arbitrarily large. Numerical plots of bifurcations are given.

A vertical cylinder is located in a free-surface wave, and a two-camera version of high-image-density particle image velocimetry is employed to characterize the spanwise modes of the flow structure in terms of instantaneous velocity and vorticity. These modes are classified according to organized patterns of velocity in the near wake, and are further interpreted in terms of distinctive arrangements of streamwise vorticity concentrations.

At low Keulegan–Carpenter number, which corresponds to small wave height, locally two-dimensional vortices having small scale and circulation tend to form as a symmetrical pair and remain attached, or in close proximity, to the surface of the cylinder. Along the span of the cylinder, the near wake shows either a sinuous S or a unidirectional U mode. The spanwise wavelength λ of the S modes, relative to the cylinder diameter D, lies in the range 1 [lsim ] λ/D [lsim ] 4:5. These values of λ/D represent the spacing between extrema of patterns of cross flow velocity, as well as between clusters of streamwise vorticity of like sign. As the free surface is approached, the value of λ/D scales with the ratio of the minor to major axes of the elliptical particle trajectory of the wave.

At moderate values of the Keulegan–Carpenter number, locally two-dimensional vortices having large scale and circulation are shed from the cylinder in an asymmetric arrangement. The corresponding spanwise mode represents the phase variation of this shedding along the span of the cylinder. These sinuous S modes involve large-scale distortions of patterns of both cross flow velocity and streamwise vorticity, which have wavelengths in the range 10 [lsim ] λ/D [lsim ] 110, in contrast to the spacing between individual concentrations of vorticity, which is 1:5D to 4D. Remarkably, it is possible to attain a unidirectional U mode, whereby the phase of the locally two-dimensional vortex shedding is preserved along the entire extent of the cylinder. Signatures of the moments due to the transverse and in-line forces on the cylinder were acquired simultaneously with the patterns of instantaneous velocity and vorticity. Severe modulations of the moment due to the transverse force are associated with spontaneous transformations between basic forms of the sinuous S and unidirectional U modes. The overall form of the signature of the moment due to the in-line force is, however, not generally affected by the spontaneous transformation between modes, but distortion of its peaks is evident.

When a vertical solid is brought in contact with the surface of a wetting liquid, a meniscus starts rising up the solid, until it reaches its steady state. We study this dynamical phenomenon experimentally with liquids of low and high viscosity, and taking as solids either large rods or small fibres. In the inviscid limit, we show that the rising time scales as √(ρr30/σ), where ρ and σ are the density and surface tension of the wetting liquid and r0 the radius of the fibre. This characteristic time holds for small fibres, with radii smaller than the capillary length a. For large rods or planar solids, r0 is replaced by a in the expression for the rising time. In the viscous limit, the rising time scales as ηr0/σ where η is the dynamical viscosity. Again, r0 is replaced by the capillary length a for large rods.

The stability of elliptically perturbed circular vortices is investigated in a two-layer shallow-water model, with constant background rotation. The fluid is bounded above and below by rigid and at surfaces. The linear stability analysis shows that elliptical perturbations are most unstable for moderate Burger numbers and vorticity shears. Shorter waves dominate for more sheared vortices. Shallow-water and quasi-geostrophic growth rates exhibit a striking similarity, except at each end of the Burger number domain. There, cyclones (anticyclones) with finite Rossby numbers are more (less) unstable than their quasi-geostrophic counterparts. A simple model gives a first-order trend for this bias.

Nonlinear model runs with initially perturbed vortices also show the similarity between the two dynamics. In these runs, elliptically deformed vortices stabilize as stationary rotating tripoles for moderate linear instability; on the other hand, strongly unstable vortices break as dipoles. During these nonlinear processes, energy transfers indicate that barotropic instability is at least as active as the baroclinic one. For tripole formation, the modal analysis of the perturbation exhibits a dominant contribution of the original wave and of the mean flow correction. The ageostrophic and divergent parts of the flow are respectively weak and negligible. The Lighthill equation proves that few internal gravity waves are generated during tripole formation or dipolar breaking. Finally, the effects of triangular perturbations on circular vortices and the formation of quadrupoles are briefly addressed.

The natural frequencies of gravity waves in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir are approximated by neglecting the off-diagonal terms of the characteristic determinant (single-mode approximation) and invoking the known results for an aperture in a half-space (well of zero depth). A parallel argument yields the corresponding results for a two-dimensional well (a slot). Comparison with Molin's (2001) numerical results for a slot suggests that the error in the single-mode approximation is [lsim ] 1%.

The authors have previously reported a model experiment on the unsteady natural convection in a triangular domain induced by the absorption of solar radiation. This issue is reconsidered here both analytically and numerically. The present study consists of two parts: a scaling analysis and a numerical simulation. The scaling analysis for small bottom slopes reveals that a number of flow regimes are possible depending on the Rayleigh number and the relative value of certain non-dimensional parameters describing the flow. In a typical situation, the flow can be classified broadly into a conductive, a transitional or a convective regime determined merely by the Rayleigh number. Proper scales have been established to quantify the flow properties in each of these flow regimes. The numerical simulation has verified the scaling results.

Three constitutive laws (Skalak et al.'s law extended to area-compressible interfaces, Hooke's law and the Mooney–Rivlin law) commonly used to describe the mechanics of thin membranes are presented and compared. A small-deformation analysis of the tension–deformation relation for uniaxial extension and for isotropic dilatation allows us to establish a correspondence between the individual material parameters of the laws. A large-deformation analysis indicates that the Mooney–Rivlin law is strain softening, whereas the Skalak et al. law is strain hardening for any value of the membrane dilatation modulus. The large deformation of a capsule suspended in hyperbolic pure straining flow is then computed for several membrane constitutive laws. A capsule with a Mooney–Rivlin membrane bursts through the process of continuous elongation, whereas a capsule with a Skalak et al. membrane always reaches a steady state in the range of parameters considered. The small-deformation analysis of a spherical capsule embedded in a linear shear flow is modified to account for the effect of the membrane dilatation modulus.

In this paper, we study the velocity field at the density interface of a two-layer stratification system when the flow is forced at the mid-depth of the lower layer by the source–sink forcing method. It is known that, in a sufficiently strong linear stratification, the source–sink forcing in certain configurations produces a single-vortex pattern which corresponds to the lowest eigenmode of the Helmholtz equation (Kanda & Linden 2001). Two types of forcing configuration are used for the two-layer experiments: one that leads to a steady single-vortex pattern in a linear stratification, and one that results in an unsteady irregular state. Strong single-vortex patterns appear intermittently for the former configurations despite the absence of stratification at the forcing height. When the single-vortex pattern occurs at the density interface, a similar flow field extends down to the forcing height. The behaviour is explained as the coupling of the resonant eigenmode at the interface with the horizontal component of the forcing jets. The results show that stratification can organise a flow, even though it is forced by an apparently random three-dimensional forcing.

A method is described for computing the motion of bubbles through a liquid under conditions of large Reynolds and finite Weber numbers. Ellipsoidal harmonics are used to approximate the shapes of the bubbles and the flow induced by the bubbles, and a method of summing flows induced by groups of bubbles, using a fast multipole expansion technique is employed so that the computational cost increases only linearly with the number of bubbles. Several problems involving one, two and many bubbles are examined using the method. In particular, it is shown that two bubbles moving towards each other in an impurity-free, inviscid liquid touch each other in a finite time. Conditions for the bubbles to bounce in the presence of non-hydrodynamic forces and the time for bounce when these conditions are satisfied are determined. The added mass and viscous drag coefficients and aspect ratio of bubbles are determined as a function of bubble volume fraction and Weber number.

The steady response of an infinite unbroken floating ice sheet to a moving load is considered. It is assumed that the ice sheet is supported below by water of finite uniform depth. For a concentrated line load, earlier studies based on the linearization of the problem have shown that there are two ‘critical’ load speeds near which the steady deflection is unbounded. These two speeds are the speed c0 of gravity waves on shallow water and the minimum phase speed cmin. Since deflections cannot become infinite as the load speed approaches a critical speed, Nevel (1970) suggested nonlinear effects, dissipation or inhomogeneity of the ice, as possible explanations. The present study is restricted to the effects of nonlinearity when the load speed is close to cmin. A weakly nonlinear analysis, based on dynamical systems theory and on normal forms, is performed. The difference between the critical speed cmin and the load speed U is taken as the bifurcation parameter. The resulting normal form reduces at leading order to a forced nonlinear Schrödinger equation, which can be integrated exactly. It is shown that the water depth plays a role in the effects of nonlinearity. For large enough water depths, ice deflections in the form of solitary waves exist for all speeds up to (and including) cmin. For small enough water depths, steady bounded deflections exist only for speeds up to U*, with U* < cmin. The weakly nonlinear results are validated by comparison with numerical results based on the full governing equations. The model is validated by comparison with experimental results in Antarctica (deep water) and in a lake in Japan (relatively shallow water). Finally, nonlinear effects are compared with dissipation effects. Our main conclusion is that nonlinear effects play a role in the response of a floating ice plate to a load moving at a speed slightly smaller than cmin. In deep water, they are a possible explanation for the persistence of bounded ice deflections for load speeds up to cmin. In shallow water, there seems to be an apparent contradiction, since bounded ice deflections have been observed for speeds up to cmin while the theoretical results predict bounded ice deflection only for speeds up to U* < cmin. But in practice the value of U* is so close to the value of cmin that it is difficult to distinguish between these two values.

The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.

The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

When a sphere settles through the free surface of a viscous fluid, the interface is deformed and assumes a funnel shape behind the sphere. If the fluid is viscoelastic and the settling process is fast compared to the relaxation time of the fluid, elastic effects are dominant and an instability occurs. The interface loses its original axisymmetry and buckles, leading to a particular mode of pinch-off unseen in Newtonian fluids. We present experimental evidence that stress boundary layers form in this type of flow, and argue that a physical mechanism for this instability can be recovered, at least qualitatively, by considering the stability of a stretched anisotropic elastic membrane in a pressure field.

The topology of the large-scale structure in the similarity region of a turbulent planar jet is investigated experimentally. The large-scale structure is reconstructed in physical space by projection of measured proper orthogonal decomposition eigenmodes onto instantaneous flow-field realizations. The instantaneous flow-field realizations are obtained by a spanwise aligned triple X-wire rake arrangement which is used in conjunction with the linear stochastic estimation technique. Instantaneous realizations are also acquired via a second triple rake arrangement which provides an assessment of the effect of spatial aliasing on the resulting structural topology. Results indicate that the self-similar large-scale structure in the planar jet consists of a dominant planar component consisting of two lines of large-scale spanwise vortices arranged approximately asymmetrically with respect to the jet centreline. This planar component of the structure resembles the classic Kármán vortex street. There is a strong interaction between structures on opposite sides of the jet in the form of nearly two-dimensional lateral streaming motions that extend well across the flow. In addition, results indicate that the effect of the nonplanar spanwise modes is to both tilt and bend the primary spanwise vortex tubes and thereby redistribute large-scale vorticity. The bending occurs primarily in the streamwise direction. The degree to which the spanwise vortices are distorted varies greatly; in some cases they are nearly streamwise oriented and in others only slight distortion of a spanwise vortex is noted. Based upon the experimental results, prospects for low-order modelling of the jet large-scale structure are discussed.

To understand how viscous and elastic membrane forces mediate the adhesion of fluid-borne cells to biological surfaces under the action of specific receptor–ligand bonds, we consider a model problem in which a two-dimensional cell interacts with a plane adhesive surface. The cell is modelled as an extensible membrane under tension containing fluid of constant volume. Assuming rapid binding kinetics, molecular binding forces are described through a contact potential that is long-range attractive but short-range repulsive. Using lubrication theory to describe the thin-film flow between the cell and the plane, we model sedimentation of the cell onto the plane under adhesive forces, followed by removal of the cell from the plane under the action of an external force. Numerical simulations show how these events are dominated respectively by quasi-steady spreading and peeling motions, which we capture using an asymptotic analysis. The analysis is extended to model a cell tank-treading over an adhesive wall in an external shear flow. The relation between cell rolling speed and shear rate is determined: at low speeds it is linear and independent of the viscosity of the suspending fluid; at higher speeds it is nonlinear and viscosity-dependent.